Pythagorist
Numbers

Cube Root Calculator

Find the cube root of any number with step-by-step working. Recognises perfect cubes, handles negative numbers (real cube roots), and shows the method clearly.

∛27

3

How it works

  1. Set up the cube root

    273\sqrt[3]{27}

    The cube root of 27 is the number which, when cubed (raised to the power 3), equals 27.

  2. Perfect cube

    33=3×3×3=273^3 = 3 \times 3 \times 3 = 27

    27 is a perfect cube. ∛27 = 3 exactly.

    Result:3

The formula

n3\sqrt[3]{n}
nnThe number (can be negative for real cube roots)

Example substitution

−273=−3since(−3)3=−27\sqrt[3]{-27} = -3 \quad \text{since} \quad (-3)^3 = -27

Negative numbers have real cube roots. −3 × −3 × −3 = −27.

Worked examples

Find the cube root of 64.

  1. 1
    Perfect cube?: 4³ = 64 ✓
  2. 2
    Result: ∛64 = 4
Answer: 4

Find the cube root of −125.

  1. 1
    Sign: Negative cube root is negative
  2. 2
    Perfect cube?: 5³ = 125 ✓
  3. 3
    Result: ∛(−125) = −5
Answer: −5

Find the cube root of 20.

  1. 1
    Perfect cube?: No (2³=8, 3³=27)
  2. 2
    Calculate: 20^(1/3) ≈ 2.7144
Answer: ≈ 2.7144

Frequently asked questions

What is a cube root?+

The cube root of n is the value that, when cubed (multiplied by itself twice), gives n. Written as ∛n.

What is ∛27?+

∛27 = 3, because 3 × 3 × 3 = 27.

What are perfect cubes?+

Numbers whose cube roots are whole numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Can numbers have negative cube roots?+

Yes. ∛(−8) = −2, because (−2)³ = −8. Unlike square roots, cube roots of negative numbers are real.

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